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Wilson Loops in the Large N Limit at Finite Temperature

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A. Brandhuber, N. Itzhaki, J. Sonnenschein, S. Yankielowicz

High Energy Physics - Theory · Submitted 17 Mar 1998 · 34 references

Pith's verdict: The paper that first put a heavy quark pair in the N=4 plasma and got thermal screening out of it; the computation is clean and the qualitative answer has held up.

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arXiv IDhep-th/9803137 (version 2)
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CitationA. Brandhuber, J. Sonnenschein, N. Itzhaki, S. Yankielowicz. "Wilson Loops in the Large N Limit at Finite Temperature." arXiv:hep-th/9803137 (1998). https://pith.science/paper/DQCOEHDL
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The paper

The paper claims

A quark and its antiparticle, held in a hot strongly coupled plasma, feel a Coulomb force at short range and essentially no force at all once their separation exceeds a distance set by the inverse temperature.

The reading

The paper asks what becomes of the force between a heavy quark and its antiparticle when the strongly coupled quantum field theory they inhabit — maximally supersymmetric $U(N)$ Yang-Mills in four dimensions — is heated. At zero temperature, in the large-$N$ limit, that force is a Coulomb law; the authors want to know how a thermal bath modifies it. Using the duality between this theory and a string moving in a curved five-dimensional geometry containing a black hole, they compute the static energy $E(L,T)$ as a function of the pair's separation $L$ and the temperature $T$. The answer is that the Coulomb law survives at short distances, with a correction of order $(TL)^4$, and that beyond a separation of order $1/T$ the energy falls to zero, so the plasma screens the pair completely. If correct, this says that a conformal gauge theory on ordinary non-compact space has no confinement/deconfinement transition at all: it simply crosses over from a Coulomb phase to a screened phase as the separation grows.

What carries the argument

The machinery is the Nambu-Goto action of a string in the near-extremal D3-brane metric, $ds^2 = (U^2/R^2)(-f(U)dt^2 + dx_i^2) + R^2 f(U)^{-1} dU^2/U^2 + R^2 dOmega_5^2$ with $f(U)=1-U_T^4/U^4$, together with the conserved 'Hamiltonian' in the $x$ direction that follows from translation invariance, $ (U^4-U_T^4)/sqrt((partial_x U)^2 + (U^4-U_T^4)/R^4) = R^2/sqrt(U_0^4-U_T^4)$. That constant of motion converts the problem into two quadratures: one relating the turning point $U_0$ to $L$ through the parameter $epsilon = 1 - U_T^4/U_0^4$, and one giving the energy after subtracting the mass of a straight string running from the boundary to the horizon. The horizon endpoint is what makes the subtraction finite and is therefore the load-bearing structural choice of the whole calculation.

Extended reading notes

Core claim

The authors take the near-extremal D3-brane supergravity solution — a black hole in the five-dimensional geometry dual to the four-dimensional gauge theory at temperature $T$ — and put a Nambu-Goto string in it, with the string's two ends on the boundary representing the quark and antiquark. Because the metric does not depend on the spatial coordinate along the pair, the string action has a conserved quantity, and the turning point $U_0$ of the hanging string is related to the separation $L$ by a single integral; the static energy is a second integral over the same variable. The new ingredient relative to the zero-temperature treatment is where the string is allowed to end: the authors argue that in the presence of a horizon the string terminates at $U=U_T$, not at $U=0$, and it is this choice that makes the mass subtraction finite. Eliminating $U_0$ between the two integrals gives a curve with a maximum separation $L_{max}$ and, before that maximum is reached, a critical separation $L_C$ at which the connected string's energy equals that of two independent straight strings. Beyond $L_C$ the connected configuration is no longer the lowest-energy one, so the physically relevant answer is two free quarks with no force between them. The resulting picture is a Coulomb-like energy $E=-2^{3/2}(4g^2_{YM}N)^{1/2}/(4^{1/4}L)$ at small $TL$, corrected by a positive constant times $(TL)^4$, and vanishing energy at large $TL$.

Load-bearing premise

The whole finite answer rests on the claim that the string dual to a heavy quark ends on the black-hole horizon rather than at the singularity, a choice the authors support with three qualitative arguments — a coordinate transformation, the fact that the Euclidean continuation covers only the exterior, and a Hawking-radiation burning estimate — rather than a derivation, and a different endpoint would change the subtraction and with it the entire energy curve.

Editorial extensions

If this is right

  • For separations much smaller than $1/T$ the quark-antiquark energy stays Coulomb-like, $E \sim -1/L$, with a correction of relative size $(TL)^4$; the combination $EL$ can depend on temperature only through $TL$, which is the theory's conformal symmetry showing itself.
  • For separations of order $1/T$ and beyond, the energy goes to zero: the thermal bath screens the pair, and two isolated free quarks become the energetically preferred configuration.
  • Because the connected string stops being the ground state at $L_C$, the calculation should be trusted only up to $L_C$, which lies below the maximum separation $L_{max}$ that the parametric curve would otherwise allow; the apparent double-valued region is unphysical.
  • On non-compact space a conformal theory has no phase transition at finite temperature, only a crossover; a transition can appear only when some scale, such as the radius of a sphere the theory lives on, breaks conformal invariance.
  • The same supergravity prescription should extend to non-conformal theories with sixteen supercharges, where a length scale is present and genuine phase transitions become possible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The screening length extracted here, $L_C T$ of order one, is a strong-coupling prediction that has no weak-coupling analogue; comparing its precise numerical value against resummed perturbative or lattice treatments of the same theory would be the sharpest test of the duality in a thermal setting.
  • The paper's subtraction scheme effectively treats the horizon as the floor of the geometry rather than the singularity, which suggests that the correct field-theory interpretation of the interior region is that it is simply absent from the gauge theory's description, not merely hard to access.
  • Because the energy vanishes smoothly rather than jumping, the crossover resembles Debye screening in an ordinary plasma; it would be natural to ask whether a screening mass can be read off the long-distance behaviour of the finite-temperature Wilson loop and matched to a thermal mass computed independently.
  • The result is stated for infinitely heavy external quarks; a finite-mass quark would introduce a string that can break, and the crossover to free quarks might then be softened or shifted, which is a computable extension within the same setup.
Original abstract

Using a proposal of Maldacena we compute in the framework of the supergravity description of N coincident D3 branes the energy of a quark anti-quark pair in the large N limit of U(N) N=4 SYM in four dimensions at finite temperature.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 1,686 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Using a proposal of Maldacena we compute in the framework of the supergravity description of N coincident D3 branes the energy of a quark anti-quark pair in the large N limit of U(N) N=4 SYM in four dimensions at finite temperature.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Using a proposal of Maldacena we compute in the framework of the supergravity description of N coincident D3 branes the energy of a quark anti-quark pair in the large N limit of U(N) N=4 SYM in four dimensions at finite temperature. contribution 0.98 0.94 0.24 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.62.

What was examined, and what was not

sources read
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  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 2. The energy of the quark-anti￾quark pair relative to the “free” quark situation as a function of ǫ = ǫ(T, U˜). Note that E = 0 is achieved at ǫC < ǫmax corresponding to LC < Lmax. For small temperature the behavior is roughly E ∼ −1/L as in the zero temperature case [31]. Taking into account the lowest corrections in U0 the expression is E = − 2 √ 2π 3/2 (4πg2 Y MN) 1/2 Γ(1/4)4 1 L [1 + c(T L) 4 ] (8) where c is a positive numerical constant which does not depend on R. The underlying conformal nature of the theory reveals itself in the fact that EL can depend on T only th
Figure 3. The energeti￾cally favorable configura￾tion for L < LC
Figure 5. The energy E of the quark-anti-quark pair as a function of L for a given T. The solid line corresponds to the numerical calculation up to LC, the dashed line indicates the expected behavior for large L. The non-conformal theories, studied in [7] from the supergravity point of view, contain 4

Showing 3 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-th/9803137 by the authors.

References
How later papers used it

73 later papers cite it

Evidence: substantial. 72 of the citing papers share no author with it, and 17 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 124 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 3 of 7 papers with this label
Compares with it 1 of 10 papers with this label
How the literature uses it

70 arXiv papers cite this in 121 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 65 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 6 papers
  • In [ref] , we computed $⟨W_0⟩$ in both extremal and non-extremal geometries following the procedure of [ref] - [ref] .

    Non extremal geometries and holographic phase transitions · Phase Transitions · 0.89

  • We follow the holographic prescription proposed in [ref] (see also [ref] and references therein for more recent discussions) to evaluate the rectangular loops in SYM in the strong t'Hooft coupling limit, $λ≫1$, with a large number of colors, $N →∞$, in terms of a classical Nambu-Goto action in the background discussed in the previous section.

    Anisotropic heavy quark potential in strongly-coupled $\mathcal{N}=4$ SYM in a magnetic field · Holographic Wilson loop $∥ℬ̌$ at $T=0$ · 1.00

  • The second contribution on the r.h.s. of [ref] is the required counter-term which subtracts the very massive quark and antiquark contributions to the regularized potential and which takes in this case the following form ($ℓ=2r+2TT≫r≃2T$ and $M=U_max-U_T/2π$): [equation] Indeed, the "W-boson string" corresponds here to an open string stretched between the brane at $U_max$ and the Schwarzschild horizon $U=U_T$ [ref] .

    Hadron potentials within the gauge/string correspondence · The conformal behaviour of the Wilson loop at finite temperature · 0.88

Compares with it 9 papers
Cites it as background 32 papers

Discussion

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